Space Time Equations

First Draft

This page has some discussion about relativity and the concepts in the lessons that was too complicated to put in a lesson.

Constant Speed of Light

The basic experimental fact that lead to Einstein's relativity is that the speed of light is constant. This was based on experimental evidence from the famous Michelson Morley experiment. This experiment measured the difference between the speed of light in several different directions. They found that no matter which directions they measured, there was no difference in the speed of light.

Lorentz Transformation

In classical Newtonian mechanics, if observer B is moving relative to A with velocity $v$

, we see

. If a third object is moving with respect to $B$

with speed $s$

, then we find its velocity with respect to $A$

. Nothing can have the same velocity with respect to $A$

and

at the same time in classical mechanics.

The Lorentz transformation says when we change our viewpoint from $A$ to $B$ , both the $x$ and $t$ coordinates change:

$x_B = \gamma(x_A + v t_A)$

$t_B = \gamma(t_A + v x_A)$
This scaling factor $\gamma$ is called the Lorentz factor, and depends on $v$

and the speed of light, $c$

:
$\gamma = \frac{1}{\sqrt{1- v^2/c^2}}$
With the Lorentz transformation, if a third object is moving with respect to $B$

with speed $s$

, its velocity with respect to $A$

is not

. In fact, if $s = c$

, then the velocity will be $c$

with respect to any other object, too.

You should visit the Wikipedia page on Lorentz Transformation or read a good physics book for more information.

Same Time or Same Place?

In classical mechanics, if two observers are moving relative to each other, $x_A = x_B + vt$

, It's almost obvious that something that is stationary with respect to $A$

is not stationary with respect to $B$

. That means the set of events that have the same $x$

coordinates for $A$

will not have the same $x$

coordinates for $B$

This is also true in Einstein's Relativity, $x_B = \gamma(x_A + v t_A)$ . However, the same is also true for the $t$ coordinate: $t_B = \gamma(t_A + v x_A)$ . The set of events that have the same $t$ coordinates for $A$ will not have the same $x$ coordinates for $B$ . That means that an "instant" for $A$ is not the same as an instant for $B$ .

Averaging Velocity

There seems to be a paradox when we try to find an average velocity. Think of a boat that travels up a river for a fixed distance, and then turns around and travels down the river. Assume that the boat's speed relative to the river is constant. That means if the speed of the river is $r$

, the speed of the boat will be $s_d = s + r$

going downstream and $s_u = s - r$

going up stream. At first thought, it seems like the total time that the boat takes should be the same no matter what $r$

is, because the average of $s_u$

and

, the speed of the boat. But this is not true, because the boat is not traveling at speed $s_d$

and

for the same amount of time. The time traveled upstream is
$T_u = \frac{L}{s_u} = \frac{L}{s-r}$
The time traveled downstream is
$T_d = \frac{L}{s_d} = \frac{L}{s+r}$
If this is confusing, you should plug in some numbers until you believe that
$T_u + T_d \ge \frac{2L}{s}$
and that they are only equal when $r=0$

. If you're daring, you can use a little Calculus to prove it.